The first example shown in Figure 2 represents sections
through a 3-D traveltime field
for a constant velocity of 2 km/s.
The grid is evenly sampled
laterally and in depth, with a sample interval of 10 m.
The source is situated at the surface, in the middle of
the plane. Figure 2a and 2b represent vertical orthogonal planes
passing through the source. The vertical plane in
Figure 2d is offset with a distance
(X=300 m) from the source. Figure 2c shows the values
of the traveltime field on a
horizontal surface passing through the middle of the cube.
Figure 3 displays the difference between the analytical solution
and the computed traveltime field for each panel shown in Figure 2.
The distribution of the errors is shown through lines of
constant value. The errors accumulate with distance from the source.
The values of the absolute error on each surface
(of order 10^{-6}) are
shown in Figure 4.

Figure 1

Figure 2

Figure 3

The constant velocity model is a poor example to examine a
finite-difference algorithm based on the
Engquist-Osher scheme, as the functions and
are zero on spherical shells of constant
radius *r*. Therefore the errors in the constant velocity case
are merely due to numerical integration and interpolation
from cartesian to spherical coordinates.

Figure 5 shows results for a traveltime field in a depth variable velocity medium. the computations are done on the same grid with a spacing of 10 m. The three panels shown in Figures 5a, 5b and 5c represent vertical planes situated at increasing distance from the source. Figure 5d displays the depth variable velocity model. The 3-D algorithm is an order of magnitude more unstable than the 2-D algorithm for comparable grid parameters in spherical, and cylindrical coordinates respectively. The instability problems disappear for smaller grid intervals in spherical coordinates, but one is limited by the amount of computer memory which can be allocated for an increasing number of grid points. The absolute error is not negligible at the sides of the panel, as can be seen from Figure 6c and 6d. Maximum values for the error appear on the edges of the model studied, at higher values of the traveltime field. The traveltimes in the 3-D model tend to be higher, as the difference between the 3-D section and the 2-D section is always positive.

Figure 4

Figure 5

12/18/1997